Branch and Bound See Beale paper

Example: Maximize z=x1+x2 x2 x1

Solve First LP problem: Solution is [ ] x2 x1

[ ] X1 <= 1 [1 1.5] x2 x1 X1 >= 2 [2 1.5], z=3.5, z=2.5

[ ] X1 <= 1 [1 1.5] x2 x1 X1 >= 2 [2 1.5], z=3.5, z=2.5 X2 = 2 No solution [2.25, 1], z=3.25

[1 1.5] x2 x1 [2 1.5], z=3.5, z=2.5 X2 = 2 No solution [2.25, 1], z=3.25 X1 = 3 [2,1], z=3 No solution

Example: Maximize x1+x2 x2 x1

S Sbar Sums edges out of S >= 2

In TSP, we solve LP problem with constraint {each vertex has 2 edges incident to it} and we add just relevant ‘subtour inequalities’ to cut off any subtour solutions. So each time we solve LP and if we get a subtour solution, we add the specific subtour inequality to cut off that solution and resolve LP. This continues until we get a final tour solution.

Objective Function to be minimized Unbounded

Infeasible solution

LP feasible, but integer infeasible